Note: Intro -- Preface -- Contents -- Hybrid Discretizations in Solid Mechanics for Non-linear and Non-smooth Problems -- 1 Introduction -- 2 Continuous, Discontinuous and Hybrid Discretizations -- 2.1 A Weakly Conforming Method -- 2.2 Incomplete Interior Penalty Galerkin Method -- 2.3 Hybrid Discontinuous Galerkin Method -- 2.4 Symmetric Hybrid Discontinuous Galerkin Method -- 2.5 Cohesive Discontinuous Galerkin Method -- 3 Numerical Evaluation of the Numerical Schemes -- 3.1 An Illustrating Smooth Example in 2D -- 3.2 A Corner Singularity in 3D -- 3.3 Cook's Membrane: A Benchmark Problem -- 3.4 Elasto-Plastic Deformation of an Annulus -- 3.5 Material Discontinuities at Interfaces: A Ring with Different Materials -- 3.6 A Fiber Composite with Nearly Incompressible Inclusions -- 3.7 A Benchmark Configuration for Thin Structures -- 3.8 An Inelastic Model Combining Plasticity and Damage -- 3.9 A Hybrid Approximation of a Contact Problem -- 4 Conclusion -- References -- Novel Finite Elements - Mixed, Hybrid and Virtual Element Formulations at Finite Strains for 3D Applications -- 1 Introduction and State of the Art -- 2 Brief Continuum-Mechanical Background -- 3 Mixed FE Technology for Large Deformations -- 3.1 Consistent Stabilization for Displacement-Pressure Elements -- 3.2 Hellinger-Reissner Principle for Large Deformations -- 4 Virtual Element Technology for Large Deformations -- 4.1 Displacement VEM Space and Projector Operators -- 4.2 Construction of Displacement Based and Two-Field Mixed VEM Approximation -- 4.3 Numerical Example -- 5 Conclusion and Outlook -- References -- Robust and Efficient Finite Element Discretizations for Higher-Order Gradient Formulations -- 1 Introduction -- 1.1 Definitions -- 2 Formulation for Finite Strain Gradient Elasticity -- 2.1 Gradient Elasticity Fundamentals -- 2.2 Rot-Free Finite Element Formulation. , 2.3 Finite Element Discretization -- 2.4 Numerical Examples -- 3 Gradient Enhanced Damage at Finite Strains -- 3.1 Continuum Damage Mechanics with Gradient Enhancement -- 3.2 Finite Elements for Gradient Damage -- 3.3 Numerical Examples -- 4 Conclusion -- References -- Stress Equilibration for Hyperelastic Models -- 1 Introduction -- 2 Hyperelasticity and Stress Equilibration -- 3 Localized Stress Equilibration -- 4 Error Estimation -- 5 Computational Experiments -- References -- Adaptive Least-Squares, Discontinuous Petrov-Galerkin, and Hybrid High-Order Methods -- 1 Introduction -- 1.1 Motivation -- 1.2 Three Nonstandard Discretizations -- 1.3 Adaptive Mesh-Refinement -- 1.4 Outline of the Presentation -- 2 Notation -- 3 Least-Squares Finite Element Methods in Computational Mechanics -- 3.1 Least-Squares Finite Element Methods -- 3.2 Natural Adaptive Mesh-Refinement -- 3.3 Alternative A posteriori Error Control -- 3.4 Axioms of Adaptivity -- 4 Least-Squares Finite Element Methods in Nonlinear Computational Mechanics -- 4.1 Convex Energy Minimization -- 4.2 Least-Squares Formulation -- 4.3 Numerical Experiments -- 4.4 Comments -- 5 Discontinuous Petrov-Galerkin -- 5.1 Optimal Test Functions -- 5.2 Breaking Spaces and Forms -- 5.3 Adaptive Mesh-Refinement -- 5.4 Axioms of Adaptivity -- 6 Discontinuous Petrov-Galerkin in Nonlinear Computational Mechanics -- 6.1 Nonlinear Discontinuous Petrov-Galerkin -- 6.2 Alternative Formulations -- 6.3 Existence and Uniqueness of Discrete Solutions -- 6.4 Numerical Experiments -- 7 Hybrid High-Order Method -- 7.1 Discrete Ansatz Space -- 7.2 Reconstruction Operators and Stabilization -- 7.3 HHO in Computational Mechanics -- 7.4 Reliable and Efficient Error Control -- 7.5 Numerical Experiment on L-Shaped Domain with Corner Singularity -- 8 HHO in Nonlinear Computational Mechanics. , 8.1 A Class of Degenerate Convex Minimization Problems -- 8.2 The Unstabilized HHO Method -- 8.3 A priori Analysis -- 8.4 A posteriori Analysis -- 8.5 A Topology Optimization Problem: Optimal Design -- References -- Least-Squares Finite Element Formulation for Finite Strain Elasto-Plasticity -- 1 Introduction -- 2 Elasto-Plasticity for the Framework of Finite Strains -- 3 Least-Squares Finite Element Formulation for Finite Strain Elasto-Plasticity -- 4 The Least-Squares Functional as an Error Estimator -- 5 Numerical Analysis -- 6 Conclusion -- References -- Hybrid Mixed Finite Element Formulations Based on a Least-Squares Approach -- 1 Introduction -- 2 Continuous Least-Squares Finite Element Formulation -- 3 Hybrid Mixed Finite Element Based on a Least-Squares Approach -- 3.1 Weak Form and Linearization of the Hybrid Mixed Formulation -- 3.2 Discretization and Implementation Aspects -- 4 Numerical Analysis for Hybrid Mixed Formulations -- 4.1 Cook's Membrane Problem -- 4.2 Quartered Plate Example -- 5 Conclusion -- References -- Adaptive and Pressure-Robust Discretization of Incompressible Pressure-Driven Phase-Field Fracture -- 1 Introduction -- 2 Notation and Equations -- 2.1 Pressurized Phase-Field Fracture in a Displacement Formulation -- 2.2 Pressurized Phase-Field Fracture in a Mixed Formulation -- 3 Discrete Formulation -- 4 Residual-Type a Posteriori Error Estimator -- 5 Numerical Tests -- 5.1 Sneddon-Inspired Test Cases (Example 1) -- 5.2 Incompressible Material Surrounded with a Compressible Layer (Example 2) -- 5.3 Nonhomogeneous Pressure Test Case with a Compressible Layer (Example 3) -- 6 Conclusions -- References -- A Phase-Field Approach to Pneumatic Fracture -- 1 Introduction -- 2 Phase-Field Model -- 2.1 Linear Elasticity -- 2.2 Finite Elasticity -- 2.3 Discretization -- 3 Multilevel Solution Strategies. , 4 Discussions and Extensions of the Phase-Field Model -- 4.1 Influencing Parameters -- 4.2 Externally Driven Fracture -- 5 Numerical Examples -- 5.1 Conchoidal Fracture -- 5.2 Pressure Driven Crack Growth -- 6 Summary -- References -- Adaptive Isogeometric Phase-Field Modeling of Weak and Strong Discontinuities -- 1 Introduction -- 2 Local Mesh Refinement -- 2.1 Truncated Hierarchical B-Splines -- 2.2 T-Splines -- 2.3 Unstructured T-Splines -- 3 Spline-Based Analysis -- 3.1 Spectral Superiority of Splines -- 3.2 Adapted Heterogeneous Spline Spaces -- 4 Adaptive Isogeometric Analysis -- 4.1 THB-Splines or T-Splines - A Computational Comparison -- 4.2 Mesh Adaptivity for Incremental Solution Schemes -- 5 Weak and Strong Discontinuities in Solid Mechanics -- 5.1 Embedded Material Interfaces in Linear Elasticity -- 5.2 Brittle and Ductile Fracture in Homogeneous and Heterogeneous Materials -- 6 Conclusion -- References -- Phase Field Modeling of Brittle and Ductile Fracture -- 1 Phase Field Model of Brittle Fracture -- 2 Exponential Shape Functions -- 2.1 Quadratic Shape Functions -- 2.2 Exponential Shape Functions -- 2.3 3d Exponential Shape Functions -- 2.4 Numerical Test -- 2.5 Adaptive Numerical Integration -- 2.6 Blending Elements -- 2.7 Adaptive Orientation 2d -- 3 Phase Field Model for Ductile Fracture -- 3.1 Phase Field Modeling of Ductile Fracture -- 3.2 Analysis of a 1D-Bar Problem -- 3.3 Plane Strain Simulations -- 3.4 3D Simulations -- 4 Concluding Remarks -- References -- Adaptive Quadrature and Remeshing Strategies for the Finite Cell Method at Large Deformations -- 1 Introduction -- 2 The Finite Cell Method -- 3 Adaptive Quadrature Based on the Moment Fitting -- 3.1 Moment Fitting -- 3.2 Adaptive Moment Fitting -- 3.3 Numerical Examples -- 4 A Remeshing Strategy for the Finite Cell Method -- 4.1 Kinematics -- 4.2 Remeshing Procedure. , 4.3 Numerical Examples -- 5 Conclusion -- References -- The Finite Cell Method for Simulation of Additive Manufacturing -- 1 Introduction -- 2 The Finite Cell Method -- 3 The Finite Cell Method Combined with Locally-Refined p-FEM -- 4 The Finite Cell Method Combined with Locally-Refined IGA -- 5 Simulation of Additive Manufacturing -- 5.1 The Thermal Model -- 5.2 Experimental Validation -- 5.3 The Finite Cell Method in SLM Process Simulations -- 5.4 Thermo-Mechanical Part-Scale Simulation -- 6 Credits and Permissions -- References -- Error Control and Adaptivity for the Finite Cell Method -- 1 Introduction -- 2 The Finite Cell Method for the Poisson Problem -- 3 Basis Functions for Finite Cell Meshes with Hanging Nodes -- 4 Residual-Based Error Estimation for the Finite Cell Method -- 4.1 Reliability -- 4.2 Numerical Example -- 5 Dual Weighted Residual Error Estimation for the Finite Cell Method -- 5.1 Error Identity -- 5.2 Refinement Strategy -- 5.3 Numerical Example -- 6 Conclusion and Outlook -- References -- Frontiers in Mortar Methods for Isogeometric Analysis -- 1 Introduction -- 2 Coupled Simulations with Mortar Methods in HPC -- 3 Basic Equations and Isogeometric Analysis -- 4 Mortar Techniques for Isogeometric Analysis -- 4.1 Biorthogonal Splines for Isogeometric Analysis -- 4.2 Multi-patch Analysis for Kirchhoff-Love Shells -- 4.3 Weak Cn Coupling for Solids -- 4.4 Crosspoint Modification -- 4.5 Hybrid Approaches for Higher-Order Continuity Constraints -- 5 Mortar Contact Formulations for Isogeometric Analysis -- 5.1 Biorthogonal Basis Functions Applied to Contact Mechanics -- 5.2 Thermomechanical Contact Problems -- 6 Multi-dimensional Coupling -- 7 Conclusions -- References -- Collocation Methods and Beyond in Non-linear Mechanics -- 1 Introduction -- 1.1 Collocation and Isogeometric Analysis. , 1.2 Beyond Collocation in Uncertainty Quantification.

Note: Intro -- Preface -- Contents -- Fundamentals of Magneto-Electro-Mechanical Couplings: Continuum Formulations and Invariant Requirements -- 1 Introduction -- 2 Foundations of Magneto-Electro Couplings -- 2.1 Preliminaries, Definitions and Units -- 2.2 A Primer in Electrostatics -- 2.3 A Primer in Magnetostatics -- 2.4 Maxwell's Equations -- 2.5 Special Cases -- 2.6 Electromagnetic Waves in Vacuum -- 2.7 Jump Conditions Across Interfaces -- 2.8 Poynting's Theorem -- 2.9 Maxwell Stress Tensor -- 3 Thermodynamics -- 3.1 First Law of Thermodynamics, Balance of Energy -- 3.2 Second Law of Thermodynamics, Entropy Inequality -- 3.3 Thermodynamic Potentials -- 4 Rotation, Spatial Reflection, Time-Reversal -- 4.1 Lorentz Invariance -- 4.2 Galilean Transformation -- 4.3 calT-Symmetry -- 4.4 Crystal Classes and Magnetic Crystal Classes -- 5 Piezoelectricity, Piezomagnetism, Some Foundations -- 5.1 Piezoelectricity -- 5.2 Piezomagnetism -- 5.3 Magnetoelectricity -- 5.4 Anisotropic and Isotropic Tensor Functions -- 6 Summary -- References -- Ferroelectric and Ferromagnetic Phase Field Modeling -- 1 Introduction -- 2 Maxwell's Equations and Polarization -- 2.1 Electro-Statics -- 3 Magnetism -- 3.1 Magneto-Statics Review -- 4 Mechano-Statics Review -- 5 Thermodynamics of Ferroelectric and Ferromagnetic Materials -- 5.1 Ferroelectric Materials: External Mechanical, Electrical Work -- and Heat Addition -- 5.2 Balance Laws for Internal Fields -- 5.3 Phase-Field Model of Ferroelectrics -- Dissipative Evolution of Domains -- Second Law of Thermodynamics -- 5.4 Internal Energy -- 5.5 Series Expansions for the Energy Functions -- 5.6 Phase-Field Modeling of Ferromagnetics -- 5.7 Finite Element Implementation -- 5.8 Example of Strain-Mediated Multiferroic Phase-Field Modeling -- References -- Semiconductor Effects in Ferroelectrics -- 1 Introduction. , 2 Thermodynamics of a Ferroelectric -- 2.1 Material Properties, Tensors, and Summation Rules -- 2.2 The Thermodynamic Energy Approach -- 2.3 The Landau-Devonshire Polynomial Approximation -- 2.4 The Depolarizing Field -- 3 Energetics of a Semiconductor -- 3.1 The Band Structure -- 3.2 Electron Statistics -- 3.3 Semiconductors with Impurities -- 3.4 Semiconductor with both Donors and Acceptors -- 3.5 Transport of Charge Carriers -- 3.6 Metal-Semiconductor Junctions -- 3.7 Heterojunctions -- 3.8 Structural Defects -- 4 The Ferroelectric Semiconductor -- 4.1 Energy Value Considerations -- 4.2 A Joint Energy Function -- 4.3 Screening -- 4.4 Maxwell-Wagner-Relaxation -- 4.5 The Electronic Impact of Defects -- 5 Case Studies -- 5.1 The Domain Wall -- 5.2 The PTCR-Effect at the Grain Boundary -- 5.3 Magnetoelectric Composites -- 5.4 Polarization Stability in Heterostructures -- 6 Conclusion and Outlook -- References -- Electromechanical Models of Ferroelectric Materials -- 1 Introduction -- 2 Origins of Ferroelectricity and Piezoelectricity -- 3 Piezoelectric Composites -- 3.1 Example of a Piezoelectric Composite -- 4 Models of Ferroelectric Switching -- 4.1 Classical Plasticity Model -- 4.2 Crystal Plasticity Model -- 4.3 Example of Crystal Plasticity Model -- 5 Models of Ferroelectric Domain Patterns -- 5.1 Theory of Compatibility -- 5.2 Average Compatibility -- 5.3 Exact Compatibility -- 5.4 Examples of Compatible Laminates -- 5.5 Evolution of Laminate Domain Patterns -- 5.6 Example of Domain Pattern Evolution -- 6 Summary and Outlook -- References -- An FE2-Scheme for Magneto-Electro-Mechanically Coupled Boundary Value Problems -- 1 Introduction -- 2 Theory of the Two-Scale Homogenization Scheme -- 2.1 Boundary Value Problems and Scale Transition -- 2.2 Discretizations of the Boundary Value Problems. , 2.3 Consistent Linearization of Macroscopic Field Equations -- 3 Magneto-Electro-Mechanical Material Models -- 3.1 Linear Piezoelectric and Piezomagnetic Model -- 3.2 Nonlinear Electrostrictive Model -- 3.3 Piezoelectric Model with Tetragonal Symmetry -- 4 Numerical Examples -- 4.1 Electrostrictive/Piezomagnetic Cantilever Beam -- 4.2 Piezoelectric/Piezomagnetic Composites -- 4.3 Ferroelectric Matrix with Cylindrical Magnetic Inclusions -- 4.4 Ferroelectric Matrix with Ellipsoidal Magnetic Inclusions -- 5 Summary -- References -- Multiscale Modeling of Electroactive Polymer Composites -- 1 Introduction -- 2 Governing Equations of Electro-Elasto-Statics at Finite Strains -- 3 Electro-Elasto-Static Boundary Value Problems on the Macro- and the Micro-scale -- 3.1 Boundary Value Problem on the Macroscopic Scale -- 3.2 Definition of Macroscopic Quantities via Homogenization -- 3.3 Boundary Value Problem on the Microscopic Scale -- 3.4 Consistent Linearization of Macroscopic Field Equations -- 4 Numerical Examples -- 4.1 Determination of the Effective Response of Electroactive Polymers with Spherical and Ellipsoidal Inclusions -- 4.2 Multiscale Simulation of Electromechanical Actuator with Composite Microstructure -- 5 Summary and Outlook -- References.

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